Find the derivative of the function.f(t)=(3t2+7t−2)10f(t)=(3t^2+7t-2)^{10}f(t)=(3t2+7t−2)10

10 ( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9 10(6t+7)(3t^2+7t-2)^9 10 ( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9

Find the derivative of the function. f ( t ) = ( 3 t 2 + 7 t − 2 ) 10 f(t)=(3t^2+7t-2)^{10} f ( t ) = ( 3 t 2 + 7 t − 2 ) 10

Here we're asked to find the derivative of the function f of t equals 3t squared plus 7t minus 2, all of that raised to the power of 10. So, of course, here we are dealing with a function inside of another function. So I have that inside function 3t squared plus 7t -2 and that outside function being all of that stuff raised to the power of 10. So here we want to use the chain rule to find this derivative. So here we're gonna be finding f prime of t using the chain rule where we wanna start outside working our way inside. So we're gonna start here with that outside function, that set stuff, raised to the power of 10. We're gonna treat that stuff as though it's just a variable right now. We don't care about it until we get to our next step when we work our way inside. So taking our derivative of our outside function here, pulling that 10 out to the front using the power rule, decreasing that power by 1, this becomes 10 times all of that stuff in the middle. We're just treating it like a variable at this point. So I'm leaving it as is. 3t squared+7tminus2 contained in those parentheses, all of this raised to the power of 9 since we have used the power rule here. Now, we are not done yet. We're working our way inside now taking the derivative of that inside function. And looking at that inside function, my polynomial, 3t squared+7t minus 2. The derivative here using the power rule yet again, multiplying by that 2, decreasing that power by 1, that gives me a 6 t. And then the derivative of 7 t is just positive 7. And the derivative of that constant is 0, so we are completely done here. Now no matter what order you choose to write this in, this is a correct answer. Typically, you'll see these presented in this order with that constant coming first and then that 6 t plus 7 because it doesn't have a very high, exponent here. And then multiplying by that 3 t squared plus 7 t minus 2 to the power of 9 just by convention. No matter what order you write this in, it is still correct if you get this answer here. Now please let me know if you have any questions, and remember that practice is the most important part here when working with derivatives. So I will see you in that next practice problem.

Find the derivative of the function. f ( t ) = ( 3 t 2 + 7 t − 2 ) 10 f(t)=(3t^2+7t-2)^{10} f ( t ) = ( 3 t 2 + 7 t − 2 ) 10

10 ( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9 10(6t+7)(3t^2+7t-2)^9 10 ( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9

10 ( 6 t + 7 ) 9 10(6t+7)^9 10 ( 6 t + 7 ) 9

9 ( 6 t + 9 ) ( 3 t 2 + 7 t − 2 ) 9 9(6t+9) (3t^2+7t-2)^9 9 ( 6 t + 9 ) ( 3 t 2 + 7 t − 2 ) 9

( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9 (6t+7) (3t^2+7t-2)^9 ( 6 t + 7 ) ( 3 t 2 + 7 t − 2 ) 9

3. Techniques of Differentiation

The Chain Rule

Business Calculus

3. Techniques of Differentiation

The Chain Rule

Struggling with Business Calculus?

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Multiple Choice

Find the derivative of the function.
$f(t)=(3t^2+7t-2)^{10}$

$10(6t+7)^9$

$9(6t+9) (3t^2+7t-2)^9$

$(6t+7) (3t^2+7t-2)^9$

$10(6t+7)(3t^2+7t-2)^9$

Verified step by step guidance

Step 1: Recognize that the function f(t) = (3t^2 + 7t - 2)^10 is a composite function. To differentiate it, we will use the chain rule, which states that if f(g(t)) is a composite function, then its derivative is f'(g(t)) * g'(t).

Step 2: Let the inner function g(t) = 3t^2 + 7t - 2 and the outer function h(u) = u^10, where u = g(t). The derivative of h(u) with respect to u is h'(u) = 10u^9.

Step 3: Differentiate the inner function g(t) = 3t^2 + 7t - 2. The derivative is g'(t) = d/dt[3t^2] + d/dt[7t] - d/dt[2], which simplifies to g'(t) = 6t + 7.

Step 4: Apply the chain rule. The derivative of f(t) is f'(t) = h'(g(t)) * g'(t). Substituting h'(g(t)) = 10(3t^2 + 7t - 2)^9 and g'(t) = 6t + 7, we get f'(t) = 10(3t^2 + 7t - 2)^9 * (6t + 7).

Step 5: Simplify the expression if needed. The final derivative is f'(t) = 10(6t + 7)(3t^2 + 7t - 2)^9.

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